The first “S&P500 options.csv” contains daily data for traditional European S&P500 options from October 30, 2017 to November 30, 2017. The file contains details on calls and puts for the November 17, 2017 and December 15, 2017 contracts.
The second data set “S&P500.xlsx” contains daily realised volatility estimates (column B) sourced from the Oxford-Man Institute Realized Library – realized.oxford-man.ox.ac.uk. Column C scales the realized volatility so that it proxies close to close volatility.
QUESTION 1 (10+1+10 = 21 marks)
This question requires you to calculate part of the volatility surface on November 8, 2017 using all available strikes that range from 2500 points to 2700 points.
A) Calculate FOUR separate implied volatility (IV) smiles using all of the provided strikes for each of the FOUR contracts i.e. i) November 17, 2017 calls; ii) November 17, 2017 puts; iii) December 15, 2017 calls and iv) December 15, 2017 puts.
Use the BSM model to extract the IVs. Further, assume a continuously compounded risk free rate of 1.30% p.a, and a continuously compounded dividend yield of 1.70% p.a.
Show all your calculations in your excel spreadsheet. Label the sheet “Surface”.
(10 marks)
B) Plot the results for each of the FOUR IV smiles on the one graph.
(1 mark)
C) Write a brief one page report outlining your findings. What can you infer from the results and are they consistent with your expectations?
(10 marks)
QUESTION 2 (57 MARKS)
Assume you are a trader seeking 1 day ahead S&P500 volatility forecasts over the month of November 2017. You require 1 day ahead forecasts that are conditional on the information set available. To illustrate, the forecast of volatility for November 1, 2017 is conditional on the information available at October 31, 2017. The forecast of volatility for November 2, 2017 is conditional on the information available at November 1, 2017 etc…
You are required to evaluate the forecasting performance of four alternative approaches over November 2017: i) implied at the money volatilities extracted using the BSM; ii) the S&P500 Volatility index or VIX; iii) the Heterogeneous Autoregressive (HAR) model fit to S&P500 realised volatilities (HAR-RV) and iv) the HAR model fit to log S&P500 realised volatilities (HAR-log-RV).
A) IMPLIED VERSUS MODEL BASED VOL FORECASTING
Write a one page report that outlines the merits of volatility forecasting using implied volatility versus model based (time series) forecasts.
(6 marks)
B) IMPLIED ATM VOL FORECASTS
i) Using the same assumptions as Question 1 above (i.e risk free rate of 1.30% p.a and a dividend yield of 1.70% p.a) extract the one day ahead forecast of S&P500 volatility for each trading day in November 2017.
You are required to construct your IV forecasts using an appropriately modified version of the technique employed in Fleming et al 1995. Here your forecast should be an at the money forecast based on the nearby contract with rollover to the next contract at the appropriate point in time.
Show your workings in your excel spreadsheet and label the sheet “IV”
(8 marks)
ii) Why is rollover performed prior to expiration and how does this relate to the surface extracted in QUESTION 1 above?
(2 marks)
iii) What assumptions about volatility are being made when you implement this forecasting procedure?
(2 marks)
C) VIX FORECASTS
i) What is the VIX? Briefly comment on how the construction of the VIX differs from the estimates obtained in PART B. (This is very technical and so only a brief paragraph outlining the key features is required).
(3 marks)
ii) Obtain the relevant time series of the VIX for forecasting purposes (citing your source) and use this to construct your one day ahead forecasts of S&P500 volatility over the month of November 2017.
Save your forecasts in your spreadsheet and label the sheet “VIX”
(3 marks)
D) MODEL BASED FORECASTING USING THE HAR-RV AND HAR-LOG-RV MODELS
i) What is realised volatility and how has it been calculated?
(2 marks)
ii) Why do we need to scale the realised volatility in this setting?
(2 marks)
iii) Briefly outline the HAR-RV model of Corsi (2009) and the economic justification for the model specification.
(3 marks)
iv) Fit the HAR-RV and HAR-log-RV models using lags over 1, 5 and 22 days. Present your estimated model including standard errors and model diagnostics. Briefly comment.
Note: excel will not allow you to estimate an OLS regression used lagged regressors, and so you will need to work out how to do this elsewhere. Other packages like Eviews are suitable.
(6 marks)
v) Use both of your estimated models to generate conditional one step ahead forecasts of S&P500 volatility over the month of November 2017.
Save your forecasts in your spreadsheet and label the sheet “HAR”
(4 marks)
E) FORECAST EVALUATION
Evaluate the forecasting performance of the four alternative approaches using the two appropriate loss functions articulated in Patton (2011).
Show your calculations in your spreadsheet.
Save your results in your spreadsheet and label the sheet “Forecasts”
(8 marks)
F) Write a one page report that outlines your findings. Your answer should include possible explanations for the differences between model forecasts. Conclude your discussion with one way that the IV and model based forecasts could be improved (i.e one suggestion for the IV approach and one suggestion for the time series approach).
(8 marks)
References
Corsi, Fulvio. “A simple approximate long-memory model of realized volatility.” Journal of Financial Econometrics 7.2 (2009): 174-196.
Fleming, Jeff, Barbara Ostdiek, and Robert E. Whaley. “Predicting stock market volatility: A new measure.” Journal of Futures Markets 15.3 (1995): 265-302.
Patton, Andrew J. “Volatility forecast comparison using imperfect volatility proxies.” Journal of Econometrics 160.1 (2011): 246-256.
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